How to skew precisely a specific part of a mesh
I think that a lattice would be a great tool.  As far as aligning it exactly, well, it's difficult (requires a bit of on-paper math) but it is possible.
However, an alternate, that's easier to set up, is to just use a mesh deform modifier:

In the bottom set of arches, the mesh deformer has been shapekeyed after the bind, creating (linearly interpolated) skew.  Note that the mesh deformer is displayed only in wire, so that you can see the interior mesh.
Note the top set of arches, which shows the setup but with the mesh deformer in the basis shapekey.  Notice that the loop cuts aren't placed evenly, as a lattice would, but are placed arbitrarily, wherever I want.  If you want to using snapping to align these edges with some edge of your base mesh, that's perfectly doable.